The Wikipedia article that Anna mentioned is an excellent description of holography and I'm not going to try and compete with it, but since I'm guessing that you're not a physicist the following might help make things clearer.
When you "see" things, you see them because they enter your eye, get focussed by the lens and hit the retina. In addition you see 3D because the image recorded by your left and right eyes are slightly different, and the brain can reconstruct a 3D image from the differences.
So if you're looking at a mouse (to use the example from the Wikipedia article) it's the light reflected off the mouse that the eye uses to "see" the mouse. Suppose you could come up with some clever trick to remove the mouse but still send the light to your eyes as if it had come from the mouse. Your brain couldn't tell the difference because your eyes are still receiving the same light as when the mouse was there. This is what a hologram does.
A hologram is a pattern of light and dark areas. When you shine a laser onto a hologram the light and dark areas scatter the light by a process called diffraction. The clever bit is that the light is scattered in exactly the same way as if there were a mouse there, so your brain sees light that looks as if it has come from a mouse, so you see a mouse. It appears in 3D because the hologram scatters light differently depending on the angle you're looking at it, so your left and right eye receive differently scattered light just as they would from a real mouse.
You might think it would be tremndously difficult to make a hologram to scatter light in just the right way to make it appear as a mouse, but actually you make a hologram from a real mouse i.e. it's just a type of photograph.
It's hard to make multicoloured holograms because to "see" the hologram you have to shine a laser on it, and lasers are just a single colour. You could use three lasers, e.g. a red, green and blue laser, but annoyingly the hologram scatters different coloured light in different ways and your multicoloured hologram would be very blurred.
I hope this helps - to get any further you'll need to work through the Wikipedia article, and also understand what diffraction is.